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Tilings of Normed Spaces

Published online by Cambridge University Press:  20 November 2018

Carlo Alberto De Bernardi
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano e-mail: [email protected], [email protected]
Libor Veselý
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano e-mail: [email protected], [email protected]
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Abstract

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By a tiling of a topological linear space $X$, we mean a covering of $X$ by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaces was initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space $X$, our main results are the following.

  • (i) $X$ admits no tiling by Fréchet smooth bounded tiles.

  • (ii) If $X$ is locally uniformly rotund $\left( \text{LUR} \right)$, it does not admit any tiling by balls.

  • (iii) On the other hand, some ${{\ell }_{1}}\left( \Gamma \right)$ spaces, $\Gamma $ uncountable, do admit a tiling by pairwise disjoint $\text{LUR}$ bounded tiles.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

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